Advances in Difference Equations
Volume 2009 (2009), Article ID 134749, 36 pages
Research Article

Results and Conjectures about Order q Lyness' Difference Equation un+qun=a+un+q1++un+1 in +, with a Particular Study of the Case q=3

1UPMC Univ Paris 06, UMR 7586: Instit. Math. de Jussieu (Univ Paris 06 and CNRS), France
2USTL Université Lille 1, UMR 8524: Laboratoire Paul Painlevé (Univ Lille 1 and CNRS), France
3Equipe d'Analyse fonctionnelle, IMJ, 16 rue Clisson, 75013 Paris, France

Received 4 March 2009; Accepted 14 July 2009

Academic Editor: Istvan Gyori

Copyright © 2009 G. Bastien and M. Rogalski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study order q Lyness' difference equation in +:un+qun=a+un+q1++un+1, with a>0 and the associated dynamical system Fa in +q. We study its solutions (divergence, permanency, local stability of the equilibrium). We prove some results, about the first three invariant functions and the topological nature of the corresponding invariant sets, about the differential at the equilibrium, about the role of 2-periodic points when q is odd, about the nonexistence of some minimal periods, and so forth and discuss some problems, related to the search of common period to all solutions, or to the second and third invariants. We look at the case q=3 with new methods using new invariants for the map Fa2 and state some conjectures on the associated dynamical system in +q in more general cases.